Proof

Proof using coset definition of normality

Proof: Since is central, this means that for all , . Thus, for a fixed , the sets and are equal, because each equals the corresponding .

Proof using conjugation definition of normality

Given: A group , a central subgroup of .

To prove: For all and , we have .

Proof: Since is central, we have, by definition, that for all . Multiplying both sides on the right by , we obtain that for all . Since by assumption, and , we obtain that .

Proof using commutator definition of normality

Given: A group , a central subgroup of .

To prove: For all , , we have .

Proof: Since we have by definition. Multiplying both sides by on the right, we get (i.e., it is the identity element). Since any subgroup contains the identity element, , so Failed to parse (syntax error): ghg^{-1}{h^{-1} \in H
.

Proof using union of conjugacy classes definition of normality

Given: A group , a central subgroup of .

To prove: is a union of conjugacy classes in .

Proof: Every element of the center of forms a conjugacy class of size 1. Since comprises only central elements, it is the union of these singleton conjugacy classes.